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'''Log-linear models''' are statistical models that are used to analyze the relationships between categorical variables. They are a type of [[Generalized linear model|generalized linear model]] that is particularly useful for examining the patterns of [[Frequency distribution|frequency distributions]] in [[Contingency table|contingency tables]]. Log-linear models can be applied in various fields, including [[Biostatistics|biostatistics]], [[Social sciences|social sciences]], and [[Market research|market research]], to explore how different factors interact with each other.

==Overview==
A log-linear model expresses the logarithm of expected cell frequencies in a contingency table as a linear function of parameters. These models are called "log-linear" because they model the natural logarithm of the expected frequencies. The general form of a log-linear model for a two-way table is:

\[
\log(\mu_{ij}) = \lambda + \lambda_i^A + \lambda_j^B + \lambda_{ij}^{AB},
\]

where \(\mu_{ij}\) is the expected frequency for cell \((i, j)\), \(\lambda\) is the overall mean effect, \(\lambda_i^A\) and \(\lambda_j^B\) are the effects of the individual factors (A and B, respectively), and \(\lambda_{ij}^{AB}\) is the interaction effect between factors A and B.

==Applications==
Log-linear models are widely used to analyze data in the form of contingency tables. Some common applications include:

* [[Epidemiology|Epidemiological studies]] to explore the relationship between diseases and potential risk factors.
* [[Sociology|Sociological research]] to study the association between various social factors.
* [[Market research]] to understand consumer behavior and preferences.

==Model Fitting==
Fitting a log-linear model typically involves estimating the parameters that best describe the observed data. This is usually done through [[Maximum likelihood estimation|maximum likelihood estimation (MLE)]]. The [[Likelihood function|likelihood function]] for a log-linear model is a function of the parameters that measures the probability of observing the given data. The MLE process finds the parameter values that maximize this likelihood function.

==Hypothesis Testing==
In the context of log-linear models, hypothesis testing is often used to assess the significance of factors and their interactions. This involves comparing a full model, which includes all terms, to a reduced model that omits some terms. The [[Likelihood ratio test]] is commonly used for this purpose.

==Advantages and Limitations==
Log-linear models offer several advantages, including the ability to handle multi-way tables and to assess interactions between factors. However, they also have limitations. For instance, they require sufficient data to provide reliable estimates, and interpreting the results can be complex, especially in models with many factors and interactions.

==See Also==
* [[Multinomial distribution]]
* [[Poisson regression]]
* [[Chi-squared test]]
* [[Factor analysis]]

==References==
<references/>

[[Category:Statistical models]]
[[Category:Categorical data analysis]]
{{Statistics-stub}}

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